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The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , where $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] , $\mathbb{R}_{0}^{+}=(0,+\infty )$ R 0 + = ( 0 , + ∞ ) . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

The main goal of this paper is investigating the existence of nonconstant positive steady states of a linear prey–predator cross-diffusion system with Beddington–DeAngelis and Tanner functional response. An analytical method and fixed point index theory plays a significant role in our main proofs.

We deal with a coupled system of k-Hessian equations: Sk( μ(D2u1)) = (-u1)) ∝ in B, Sk( μ(D2u1)) = (-u2)) in B, u1 < 0, u2< 0 in B, u1 = u2= 0 on ∂B, where k = 1,2, ⋯ , N, B is a unit ball in RN, N ≥ 2, α and β are positive constants. By using the fixed-point index theory in cone, we obtain the existence, uniqueness and nonexistence of radial convex solutions for some suitable constants α and β. Furthermore, by using a generalized Krein-Rutman theorem, we also obtain a necessary and sufficient existence condition of the convex solutions to a nonlinear eigenvalue problem. &nbsp

We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.

In this paper, we investigate the existence of positive solutions for a nonlocal fractional boundary value problem involving Caputo fractional derivative and nonlocal Riemann–Stieltjes integral boundary condition. By using the spectral analysis of the relevant linear operator and Gelfand’s formula, we obtain an useful upper and lower bounds for the spectral radius. Our discussion is based on the properties of the Green’s function and the fixed point index theory in cones.

In this paper, we discuss the positive solutions of beam equations with the nonlinearities including the slope and bending moment under nonlocal boundary conditions involving Stieltjes integrals. We pose some inequality conditions on nonlinearities and the spectral radius conditions on associated linear operators. These conditions mean that the nonlinearities have superlinear or sublinear growth. The existence of positive solutions is obtained by fixed point index on cones in $C^{2}[0,1]$C2[0,1], and some examples are given for beam equations subject to mixed integral and multi-point boundary conditions with sign-changing coefficients.

We investigate positive solution for high-order fractional differential equations with resonant boundary value conditions. By using the spectral theory and the fixed point index theory, both the nonexistence and existence results of the resonant boundary value problems are given.

In the present work, we have studied a diffusive tritrophic food chain model in which the species at each trophic level interact in accordance with Crowley–Martin functional response under mixed boundary conditions. Using degree theory and fixed point index-based methods, we have proved the existence of the positive solutions of the proposed system. We have proved the permanence of the positive solutions and existence of global attractor. The conditions for diffusion-driven instability have been obtained analytically. Moreover, the pattern formation due to diffusion-driven instability has been investigated numerically. We have shown the existence of the positive solutions both analytically and numerically.

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

In this article, we study a class of fractional differential equations with resonant boundary value conditions. Some sufficient conditions for the existence of positive solutions are considered by means of the spectral theory of linear operator and the fixed point index theory.

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the fixed point index theory.

In this article, we study the existence of positive solutions to a class of two-term fractional nonlocal boundary value problems. The existence and multiplicity of positive solutions are established by means of fixed point index theory. The nonlinearity f(t,x) $f(t,x)$ permits a singularity at t=0,1 $t = 0,1$ and x=0 $x=0$.

In this paper we use the fixed point index to study the existence of positive solutions for a system of nonlinear Hadamard fractional differential equations involving coupled integral boundary conditions. Here we use appropriate nonnegative matrices to depict the coupling behavior for our nonlinearities.

We investigate a class of semipositone fractional integral boundary value problem on the half-line. Using the fixed point index theorems in a cone, we obtain the existence result of positive solutions.

We prove the existence of multiple positive solutions of nonlinear second-order nonlocal boundary value problems with nonlinear term having derivative dependence. We allow the nonlinearity to grow quadratically with respect to derivatives. We obtain a priori bounds for norms of derivatives by using a recently obtained Gronwall-type inequality. Three examples illustrate some of the results. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

The purpose of this work is to establish an extension of a Bai-Ge type multiple fixed point theorem for a sum of two operators. The arguments are based upon recent fixed point index theory in cones of Banach spaces for a k-set contraction perturbed by an expansive operator. As illustration, our approach is applied to prove the existence of at least three nontrivial non-negative solutions for a class eigenvalue three-point BVPs for a class of fourth order ordinary differential equations (ODEs for short).

In this paper, using a fixed point index theorem on a cone, we present some existence results for one or multiple positive solutions to &phi, Laplacian nonlocal boundary value problems when &phi, is a sup-multiplicative-like function and the nonlinearity may not satisfy the L 1 -Carath e &acute, odory condition.

This paper deals with the study of the existence and non-existence of solutions of a three-parameter family of nonlinear fractional differential equation with mixed-integral boundary value conditions. We consider the &alpha, Riemann-Liouville fractional derivative, with &alpha, &isin, ( 1 , 2 ] . To deduce the existence and non-existence results, we first study the linear equation, by deducing the main properties of the related Green functions. We obtain the optimal set of parameters where the Green function has constant sign. After that, by means of the index theory, the nonlinear boundary value problem is studied. Some examples, at the end of the paper, are showed to illustrate the applicability of the obtained results.

We apply the fixed point index to obtain positive solutions of a nonresonant periodic boundary value problem for a third-order differential equation u‴+ρ3u=λfu.

In this paper, we establish the existence of nontrivial positive solution to the following integral-infinite point boundary-value problem involving ϕ-Laplacian operator D0+αϕx,D0+βux+fx,ux=0,x∈0,1,D0+σu0=D0+βu0=0,u1=∫01 gtutdt+∑n=1n=+∞ αnuηn, where ϕ:0,1×R→R is a continuous function and D0+p is the Riemann-Liouville derivative for p∈α,β,σ. By using some properties of fixed point index, we obtain the existence results and give an example at last.

We establish the existence, nonexistence, and multiplicity of positive solutions to semilinear elliptic systems with integral boundary conditions when positive multiparameters vary on the boundary. We prove the results by using sub- and supersolution argument and fixed point index theory.

We investigate a singular fractional differential equation with an infinite-point fractional boundary condition, where the nonlinearity $f(t,x)$ may be singular at $x = 0$ , and $g(t)$ may also have singularities at $t= 0$ or $t=1$ . We establish the existence of positive solutions using the fixed point index theory in cones.

Using the fixed point index, we establish two existence theorems for positive solutions to a system of semipositone fractional difference boundary value problems. We adopt nonnegative concave functions and nonnegative matrices to characterize the coupling behavior of our nonlinear terms.

We provide new results on the existence, non-existence and multiplicity of non-negative radial solutions for semilinear elliptic systems with Neumann boundary conditions on an annulus. Our approach is topological and relies on the classical fixed point index. We present an example to illustrate our theory.

In this paper we use the fixed point index to establish the existence of positive solutions for a system of first-order discrete fractional boundary value problems with semipositone nonlinearities. Concave and convex functions are adopted to characterize the coupling behavior of our nonlinearities.

This paper deals with the effect of parameters on properties of positive solutions and asymptotic behavior of an unstirred chemostat model with the Beddington–DeAngelis (denote by B–D) functional response under the Robin boundary condition. Firstly, we establish some a priori estimates and a sufficient condition for the existence of positive solutions (see (Feng et al. in J. Inequal. Appl. 2016(1):294, 2016)). Secondly, we study the effect of the small parameter $k_{1}$ and sufficiently large $k_{2}$ in B–D functional response, which shows that the model has at least two positive solutions. Thirdly, we investigate the case of sufficiently large $k_{1}$ . The results show that if $k_{1}$ is sufficiently large, then the positive solution of this model is determined by a limiting equation. Finally, we present an asymptotic behavior of solutions depending on time. The main methods used in this paper include the fixed point index theory, bifurcation theory, perturbation technique, comparison principle, and persistence theorem.

By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example, when dealing with boundary value problems where nonlocal terms occur in the differential equation and/or in the boundary conditions. Some examples are given to illustrate the theoretical results., 29 pages

We consider a constrained semilinear evolution inclusion of parabolic type involving an \begin{document}$m$\end{document} -dissipative linear operator and a source term of multivalued type in a Banach space and topological properties of the solution map. We establish the \begin{document}$R_δ$\end{document} -description of the set of solutions surviving in the constraining area and show a relation between the fixed point index of the Krasnosel'skii-Poincare operator of translation along trajectories associated with the problem and the appropriately defined constrained degree of the right-hand side in the equation. This provides topological tools appropriate to obtain results on the existence of periodic solutions to studied differential problems.

In this work we show the existence of coexistence states for a nonlocal elliptic system arising from the growth of cancer stem cells. For this, we use the bifurcation method and the theory of the fixed point index in cones. Moreover, in some cases we study the behaviour of the coexistence region, depending on the parameters of the problem.

We study the homogeneous Dirichlet boundary value problem of a singular $$(p_1,p_2)$$ -Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains.

In this paper we prove the existence of continua of nonradial solutions for the Lane–Emden equation in the annulus. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane–Emden problem has a very rich and complex structure.

Using the theorem and properties of the fixed point index in a Banach space and applying a new method to dispose of the impulsive term, we prove that there exists a solvable interval of positive parameter λ in which the second order impulsive singular equation has two infinite families of positive solutions. Moreover, we also establish the new expression of Green’s function for the above equation. Noticing that $\lambda>0$ and $c_{k}\neq0$ ( $k=1,2,\ldots,n$ ), our main results improve many previous results. This is probably the first time that the existence of two infinite families of positive solutions for second order impulsive singular parametric equations has been studied.

Motivated by boundary value problems we give new results for a class of nonlinear Hammerstein integral operators acting in a cone to have a fixed point index equal to one. The idea is to allow the nonlinearity to be large on one part of its domains provided it is sufficiently small on a second part. Stronger results are obtained when the nonlinearity is decreasing on the second part of its domain. This allows new classes of nonlinearities to be treated and existence of a positive solution is established under weaker conditions than in previous works. The results are flexible and are not tied to any particular boundary conditions but can be applied to very many problems. We give several examples including applications to problems arising in chemical reactor theory.

In this paper, we give improved results on the existence of positive solutions for the following one-dimensional p-Laplacian equation with nonlinear boundary conditions: $$ \textstyle\begin{cases} (\phi_{p} ( y'' )) ' + b ( t ) g ( t, y ( t ) ) = 0, \quad 0 < t < 1, \\ \lambda_{1}\phi_{p} ( y ( 0 ) ) - \beta_{1} \phi_{p} ( y' ( 0 ) ) = 0, \\ \lambda_{2}\phi_{p} ( y ( 1 ) ) + \beta_{2} \phi_{p} ( y' ( 1 ) ) = 0,\qquad y'' ( 0 ) = 0, \end{cases} $$ where $\phi_{p} ( s ) = | s | ^{ p-2 } s$ , $p >1 $ . Constructing an available integral operator and combining fixed point index theory, we establish some optimal criteria for the existence of bounded positive solutions. The interesting point of the results is that the term $b ( t ) $ may be singular at $t=0$ and/or $t=1$ . Moreover, the nonlinear term $g(t, y)$ is also allowed to have singularity at $y=0$ . In particular, our results extend and unify some known results.

This paper deals with the existence of positive radial solutions of the elliptic equation with nonlinear gradient termwhere , , is continuous. Under the conditions that the nonlinearity may be of superlinear or sublinear growth in u and , existence results of positive radial solutions are obtained. For the superlinear case, the growth of f in is restricted to quadratic growth. The superlinear and the sublinear growth of the nonlinearity of f are described by inequality conditions instead of the usual upper and lower limits conditions. Our discussion is based on the fixed point index theory in cones.

In this paper we study the existence of positive solutions for the fractional $p$-Laplacian boundary value problem \[\left\{\aligned & D_{0+}^\beta (\phi_p (D_{0+}^\alpha u(t)))=f(t,u(t)), t\in (0,1),\\ & u(0)=u'(0)=0, u'(1)=au'(\xi), D_{0+}^\alpha u(0)=0, D_{0+}^\alpha u(1)= b D_{0+}^\alpha u(\eta), \endaligned \right.\] where $2 1$, $\phi_p^{-1}=\phi_q$, $1/p+1/q=1$, $0

This article is to study a three-point boundary value problem of Hadamard fractional p-Laplacian differential equation. When our nonlinearity grows ( p &minus, 1 ) -superlinearly and ( p &minus, 1 ) -sublinearly, the existence of positive solutions is obtained via fixed point index. Moreover, using an increasing operator fixed-point theorem, the uniqueness of positive solutions and uniform convergence sequences are also established.

In this article, we consider a class of singular fractional differential equations with nonlocal boundary value conditions. The existence and multiplicity of positive solutions are derived by the fixed point index theory, and the nonlinearity $f(t,x)$ may be singular at $t=0,1$ and $x=0$ . The interesting point is that the existence results are closely associated with the relationship between 1 and the spectral radii corresponding to the relevant linear operators. An example is also given to demonstrate the validity of the main results.

In this article we study applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we present a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs are quite different, and most remarkably some of the proofs become simpler and more straightforward. As an application, we obtain the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advances the recent results obtained in several articles even in a more general setting.

In this article, we investigate the existence of positive solutions to fourth-order problems with dependence on all derivatives in nonlinearities subject to the Stieltjes integral boundary conditions $$\begin{aligned}& \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u'(0)+\beta _{1}[u]=0, \qquad u''(0)+\beta _{2}[u]=0, \qquad u(1)=\beta _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ and $$\begin{aligned}& \textstyle\begin{cases} -u^{(4)}(t)=g(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=\alpha _{1}[u], \qquad u'(0)=\alpha _{2}[u], \qquad u''(0)=\alpha _{3}[u], \qquad u'''(1)=0, \end{cases}\displaystyle \end{aligned}$$ where $f: [0,1]\times \mathbb{R}_{+}\times \mathbb{R}_{-}^{3}\to \mathbb{R}_{+}$ , $g: [0,1]\times \mathbb{R}^{3}_{+}\to \mathbb{R}_{+}$ are continuous and $\beta _{i}[u]$ , $\alpha _{i}[u]$ ( $i=1,2,3$ ) are linear functionals involving Stieltjes integrals of signed measures. Some growth conditions are posed on nonlinearities f, g, meanwhile the spectral radii of corresponding linear operators are restricted, which means the superlinear or sublinear conditions. On the cones in $C^{3}[0,1]$ we apply the theory of fixed point index, the existence of positive solutions is obtained. We also give some examples under mixed multi-point and integral boundary conditions with sign-changing coefficients.

In this paper, we study power series having a fixed point of multiplier 1. First, we give a closed formula for the residue fixed point index, in terms of the first coefficients of the power series. Then, we use this formula to study wildly ramified power series in positive characteristic. Among power series having a multiple fixed point of small multiplicity, we characterize those having the smallest possible lower ramification numbers in terms of the residue fixed point index. Furthermore, we show that these power series form a generic set, and, in the case of convergent power series, we also give an optimal lower bound for the distance to other periodic points., Comment: 29 pages; minor revision from v2

We study existence and multiplicity of positive radial solutions for a semipositone coupled elliptic system in exterior domains where the nonlinearities depend on the gradients and the Neumann boundary conditions are nonlocal. We use a nonstandard cone to establish existence of multiple solutions by means of fixed point index theory.

Consider the equation − M ( x , ‖ u ‖ ) Δ p u = λ f ( x , u , ∇ u ) − g ( x , u , ∇ u ) in Ω, u = 0 on ∂Ω. The main aim of this paper is to prove existence results for both non-degenerate and degenerate cases of the function M. Our approach relies on the fixed point index and the cone theoretic argument.

In this paper, using the theory of fixed point index, we prove some results on the multiplicity of positive solutions for a class of nonlocal elliptic problem.

We consider elliptic Dirichlet problem − Δ p u = λ f ( x , u , ∇ u ) − g ( x , u ) in Ω , u = 0 on ∂ Ω . Assume that the nonlinearity f satisfies certain growth condition. Using the fixed point index, the Leggett–Williams theorem on fixed point and the arguments on monotone minorant, we prove the existence and multiplicity results for the equation. This extends some known results.

In this paper, the existence of positive solutions for a class of fractional differential equations with Riemann–Stieltjes integral boundary conditions are investigated under some weak conditions concerning the spectral analysis of the relevant linear operator and Gelfand’s formula by means of the fixed point index theorem in cones. The nonlinearity permits singularities not only at \(t=0,1,\) but also at \(x_{i}=0\ (i=1,2,\ldots ,n-1)\). The results obtained herein generalize and improve some known results including singular and non-singular cases.